tant in the whole work, and a subject of the most difficult nature; it is, to find how much the attraction of one planet will disturb the motion of another in its orbit: and here the author has entered into a very full inveftigation of every thing relative 10 the subject, and fo clearly explained every part of this intricate process, that by an attentive reader it may be easily underftood. If r be the radius rector of the planet attracted, f the force tending to the fun, F the force acting perpendicularly to the radius rector, v the angle defcribed from any given point, in the time t; then this author has proved, that

And here we will note an erratum in the printing, not taken notice of by the author in the errata. In the fecond equation, it is printed F t'inftead of ft. From these two equations, the curve defcribed by the body attracted may be found. Now, if

➡s, we obtain s++Po, the integration of which

gives the equation of the required curve. If we put Pa' x cos. m v + b'x cos. n v +, &c. and for s we substitute 1 —

a2

Cr

and g and c be affumed the corrections of certain fluents,

-, &c. be the equation of the curve required, upon the above fuppofition, for the value of P, which fuppofition is always applicable in the cafe of the planets.

If r the radius rector of an ellipfe from the focus, p= the femiparameter, c the diftance of the focus from the centre divided by the feini-axis major, v the true anomaly,

and

and p =

a2

; then if we suppose the motion to begin from the

higher apfide, go, and we get = 1

a1

(c m2 -I

X cos. m v

n2

b'

m2 I

n2 I

x cos. n v—, &c. the equation of the curve. Or, putting u=c

a1

a1

-, &c. and x ( m2

I

X cos. mv

x cos. 'n v-, &c.)s, the equation becomes

X cos. vs, where 44 X cos. v is the equation of a new ellipfe. Hence the author concludes, that the ellipfe which would have been defcribed without the disturbing force, is changed into another ellipfe very nearly, the deviation from an ellipfe being only that which arifes from the fmall quantity S. The effect therefore of the difturbing forces is, to change the eccentricity of the ellipfe, to alter the mean distance, and to cause a small alteration of this new ellipfe. Having determined the general equation of the curve, the author next makes the application. Let us affume CI; now=P the femiparameter; then the orbit being supposed to have but a fmall eccentricity, p I, very nearly; hence we may af fume a=. Let E be the body attracted, P the attracting body, M its mafs, the fum of the maffes of E and the fun S being unity; and let E Sd, PS b, EPk, 2 angular distance of E from P at the fun; then we get Mx d /Mx b

Thefe are then the general equations, from which the equa tions arising from the difturbing force are to be found. If the body attracted be the earth, and the difturbing body be Jupiter; then the author finds the equations of the mean motion of the earth arifing from the attraction of Jupiter to be 7", 1 x fin. v + 2",7 x fin. 2 y + 0,4 × fin. y X !",5 × fin. 2 yx, where x = the mean longitude of the earth, and y = the mean longitude of the earth that of Jupiter. The author next explains the methods given by Euler and Le Grange of refolving b2+r-2 b г. cos. 2 r.

feries A+B cos. z + C cos. 2 z D,cos. 3 z +, &c. This refolution being of the first importance in phyfical astronomy, and particularly in the bufinefs of this chapter.

He next proceeds to find the effect of the difturbing force of Venus upon the earth; and putting t the mean longitude of

Venus - that of the earth, he finds the equations hence arifing to be 5",3 x fin. t 6" x fin. 2 t 07 X fin, 30′′,2 X fin. 4 t.

The next application is that of finding the motion of the moon's apogee. This is a problem of great difficulty; but the author has entered into very full examination of it, and explained at length every ftep of the procefs; the reader will therefore here find the investigation very intelligible, and the reafon of the operation rendered very clear. If at the fame time that the planet defcribes the angle v, the apfide defcribe the angle vm v, then the motion of the planet in refpect to the aplide, or the true anomaly, is m v; and the equation of the moveable ellipfe becomes-x cos. m v, r being the radius rector, d half the parameter, and w the eccentricity divided by the femi-axis major. From the general equation of the curve before given, our author deduces the following equation of the lunar orbit, =1 w cos. mv+p'cos.

COS.

n

The terms

d

d

r

W

:

2 V

D

- q'

(— — m). v + r' cos. (~ + m). v + s' cos. (+m). v + s' cos. (m2). v, where n- In :: mean motion of the fun that of the moon, and p', q', r', s', given quantities. He next proceeds to explain a point, of which we have never before seen an explanation. w cos. m v are the principal ones in the equation, and denote a moveable ellipfe, containing the great equation of the moon's motion, that is, the equation of the centre; alfo, the motion of the apogee. And as this equation does not depend upon the fituation of the fun, the motion of the apogee, which is denoted by it, may be confi dered as the mean effect of the difturbing force. This motion of the apogee is conftantly progreffive, and is in proportion to the motion of the body, as I -in: 1; if, therefore, I reprefent the mean motion of the moon, I —m will represent the mean motion of the apogee. The other terms are fmall, and depending on the pofition of the fun in refpect to the moon, they will produce fome of the fmaller equations of the moon's motion, and the equations of the motion of the apogee. Hence, we may confider — 1 — w còs. m v as an equation, reprefenting the bafis of the lunar orbit. The next operation is therefore to determine the value of m, and as the motion of

d

the apfide is flow, m is first affumed equal to 1, and then by Correction, its true value is found to be very nearly0,99164, and therefore — m = 0,00836 the motion of the apogee, the mean motion of the moon being unity; and by obfervation it is found o,cc8455. Now in finding the value of m, fome very fmall quantities were omitted; the operation therefore ought to be a very near approximation, and accordingly we find it to be fo; hence we may conclude, that the theory of gravity is fuflicient to account for the motion of the moon's apogee.

Chap. XXVIII. is upon the Tides. Tides are caufed by the attraction of the fun and moon on the waters upon the furface of the earth; the computations of the effects are therefore made upon the principle of the law of gravitation. Kepler was the first who affigned the true phyfical caufe of this phænomenon; but Newton was the first who gave the principles of the calculation. The prefent author first proves, that if the earth were a perfect fphere, and without any rotation, the figure which it would put on from the attraction of the fun or moon, would be that of a fpheroid: and, from the attraction of the fun, he computes that the difference of the radii will be 2,033 feet, and from the moon's attraction, that it will be 5,412 feet. He next proceeds to explain the method given by D. Bernoulli, who has taken for granted that the earth will put on the form of a spheroid. If the difference of the radi arifing from the fun's attraction be m, and that of the moon be n, and the fun and moon be in a meridian paffing through the pole of a fpheroid; and b be the radius of the-earth, s the cofine of the diftance of the fun from any place on the abovementioned meridian, r the cofine of the moon's diftance, then the altitude of the tide at that place will be 3r2- bz 12

3 s2 b2
3 b2

xm+

x n. From hence a method is given to find the 3 b2 ratio of m to n, which appears to be that 1: 21. The method given by Sir I. Newton is fubject to great uncertainty. A rule is next given to find the effect arifing from the declination of the moon; and in like manner for the fun: and hence the general effect of the tides on different parts of the earth, and in different fituations of the moon, is difcovered. If S = the cofine of the moon's declination, Cits fine, s the fine of the distance of the place from the pole, c = the cotine, y = the cofine of the angle between the meridians paffing through the place and the moon; then from the effect of the moon, Ssy+Ccxm the height of the water above the lowest

point. Hence this author deduces ten of the most remarkable cafes. He fhows why fmall collections of waters are not fubject to much tide; and gives two tables for finding the times and heights of the high tides. The reader will here find great fatisfaction upon this fubject.

Chap. XXXIX. is upon the Principles of Projection, and the Conftruction of geographical Maps. The author here explains the principles of orthographic, ftereographic, and Mercator's projection; and then applies them to the conftruction of the refpective maps. He points out the imperfection of thefe maps, in giving the true reprefentations of countries; and explains the particular utility of the latter conftruction, in navigation.

Chap. XL. is on the Ufe of Interpolations in Aftronomy. The author has here inveftigated the rule for interpolations, and very clearly explained the principle; and then applied it to a variety of examples. He has alfo fhown, that the rule given by Dr. Halley, for finding the time of the folftice, cannot be depended

upon.

Chap. XLI. is upon the Hiftory of Aftronomy. Here the author has traced out the rife and progrefs of aftronomy, giving an account of all the difcoveries which have been made in this branch of fcience, and to whom we are indebted for them. It is divided into the following heads: on the Aftronomy of the Egyptians and Chaldeans; on the Aftronomy of the Chinefe and Indians; on the Aftronomy of the Greeks to the Time of Ptolemy; on the Aftronomy of the Arabs, Perfians, and Tartars; on the Progrefs of Aftronomy, from its Resto ration in Europe.

The author having thus completed his valuable work, proceeds in his conclufion to take notice of thofe extraordinary. marks of defign in the conftruction of the universe, which prove fo clearly that it could not have owed its formation to chance, but to the contrivance of Infinite Wifdem. The proofs here adduced in fupport of a Deity, are of fo ftrong and fatisfactory a nature, that, to a mind open to conviction upon rational grounds, their force is little inferior to demonftration. We cannot, by abridging this part, do juftice to the author; but we earneftly recommend it to the ferious attention of the reader, as we think it cannot fail to convince him, that the fyftem of the univerfe is the work of an infinitely powerful, wife, and good Be ng. We will, however, prefent the reader with the conclufion.

"If we carry our views up to the firmament of the fixed ftars, the power of the Deity will be ftill mote aftonishing. Let a man contemplate